General Topology
Academic Year 2023/2024 - Teacher: Angelo BELLAExpected Learning Outcomes
Training in the use of formal language in abstract mathematics. The course provides a complete description of the basic facts of General Topology. Particular emphasis will be given to the discussion of examples and exercises.
At the conclusion of the course, the students shuold be able to understand the basic notions, to apply their knowledge and understanding. They also should be able to give oral and written presentation of the most important theorems of the contents of the course as well as to work both in collaboration with other people and by themselves, making judgements.
Learning assessment may also be carried out on line, should the conditions require it.
Course Structure
Lectures with slides and exercises in which the assigned exercises are corrected.
If the teaching is given in a mixed or remote way, the necessary changes may be introduced with respect to what was previously stated, in order to comply with the program envisaged and reported in the syllabus.
PLEASE NOTE: Information for students with disabilities and / or SLD
To guarantee equal opportunities and in compliance with the laws in force, interested students can ask for a personal interview in order to plan any compensatory and / or dispensatory measures, based on the didactic objectives and specific needs.
It is also possible to contact the referent teacher CInAP (Center for Active and Participatory Integration - Services for Disabilities and / or SLD) of our Department, prof. Filippo Stanco
Required Prerequisites
Attendance of Lessons
Detailed Course Content
The notion of topological space. Open and closed sets. Bases and fundamental systems of neighborhoods. Construction of a topology. First and second axioms of countability. Continuous functions and homeoformisms. Subspaces and hereditary properties. Product of topological spaces: the finite case and the general case. Quotient spaces. Metric spaces and metrizable spaces. Separation axioms. Normal spaces and Urysohn's lemma. The Tietze extension theorem. Compact spaces and their fundamental properties. Tychonoff's theorem. The embedding theorem. A fundamental characterization of complete regularity. The notion of compactification. Connected spaces and their properties. Connectidness of a product. Locally compact spaces and by Aleksandroff's compactification.
Textbook Information
1. Professor's notes.
2.Topologia by M. Manetti. General Topology by R. Engelking.
Course Planning
| Subjects | Text References | |
|---|---|---|
| 1 | La nozione di spazio topologico. | 1 |
| 2 | Insiemi aperti e chiusi. Basi e sistemi fondamentali di intorni. | 1 |
| 3 | Costruzione di una topologia. Primo e secondo assioma di numerabilità. | 1 |
| 4 | . Funzioni continue ed omeoformismi. Sottospazi e proprietà ereditarie. | 1 |
| 5 | Prodotto di spazi topologici: il caso finito e il caso generale. Spazi quoziente. Spazi metrici e spazi metrizzabili. | 1 |
| 6 | Assiomi di separazione. Spazi normali e lemma di Urysohn. Il teorema di estensione di Tietze | 1 |
| 7 | Spazi compatti e loro proprietà fondamentali. Il teorema di Tychonoff. Il teorema di immersione | 1 |
| 8 | Una caratterizzazione fondamentale della completa regolarità. La nozione di compattificazione. Spazi connessi e loro proprietà. | 1 |
| 9 | La connessione di un prodotto. Spazi loalmente compatti e compattificazione di Aleksandroff. | 1 |
Learning Assessment
Learning Assessment Procedures
Learning assessment may also be carried out on line, should the conditions require it.